[FOM] Infinitesimal calculus

joeshipman@aol.com joeshipman at aol.com
Tue May 26 20:24:34 EDT 2009

I disagree. Real numbers can be explained to high school students 
perfectly soundly in terms of successive rational approximations 
(decimal expansions, binary expansions, continued fractions, etc.) to 
points on a geometric line. When this is done, they have a consistent 
intuition for "what real numbers are", an intuition which makes a proof 
of the least upper bound property quite easy to understand. There is no 
philosophical difficulty at this level, unless you try to get across 
the point that physical space might not have a structure that is 
precisely captured by this definition.

On the other hand, when you use infinitesimals you have to be able 
either to say "what an infinitesimal is" and distinguish infinitesimals 
 from the standard reals in a way that is intuitive and not merely 
formal, or else you have to deny the Archimedean axiom and say that 
infinitesimals are just as real as standard real numbers.  This CAN be 
done in an intuitive way following Conway's "On Numbers and Games", but 
all the attempts I have seen to use infinitesimals to develop Calculus 
for students do not do this, instead  treating them purely formally in 
a system where you can't get your hands on an infinitesimal directly. 
This is unsatisfactory for high-school level students who aren't 
usually comfortable with introducing elements that can be regarded as 

(In Conway's system you just extend Dedekind cuts into the transfinite, 
so each individual number can be expressed as a map from some ordinal 
to the set {+, -} and when the ordinal is at most \omega you recover 
the classical real numbers as a substructure.)

-- JS

-----Original Message-----
From: David Ross <ross at math.hawaii.edu>

I don't really want to engage yet again in the argument as to whether 
it is
better or not to teach calculus with infinitesimals, I just want to 
out that some of the remaining arguments against doing so do not hold 
up to
strong scrutiny.  When we made the switch in the US 100 years ago, it 
for reasons of rigor; now, however, it is ultimately just a matter of 

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